Black holes are laboratories for testing fundamental theories that explain how the Universe works on the largest and the smallest scales (e.g., General relativity and Quantum physics) and a possible conciliation between them i.e quantum gravitational theory.
In fact, black hole thermodynamics plays a main role in our understanding of the mysterious nature of such compact objects [1]}, [2]}. One of the most intriguing enigmas in black hole thermodynamics is the Hawking-Page (\(HP\) ) phase transition between the stable large black hole and the thermal gas. It was first investigated for the Schwarzschild black hole in Ref. [2]} and then for the charged black hole in Refs. [4]}, [5]}, especially in the context of the AdS/CFT correspondence [6]}, [7]}, [8]}, [9]}, [10]}. After the birth of the extended phase space notion in the AdS spacetime, this area was reexamined again with prolific outcomes [11]}, [12]}, [13]}, [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}, [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}. Moreover, it's largely understood that a black hole in an asymptotically flat space has negative heat capacity and is thus thermodynamically unstable. Therefore, when a black hole radiates via the Hawking mechanism [28]}, it can no longer maintain thermal equilibrium with the environment and will evaporate eventually. In addition, the non-extensive nature of the Bekenstein-Hawking entropy of black holes which is proportional to the surface area of its event horizon rather than the volume, and in the strong gravitational scheme near the black hole, the condition of negligible long-range type interactions in the standard statistical descriptions breaks down, and consequently, the usual definition of mass and other extensive quantities is not possible locally pushes us to go beyond the standard Gibbs-Boltzmann (\(GB\) ) statistical proposal.
